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Stationary amplitudes of quantum walks on the higher-dimensional integer lattice

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Abstract

Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher-dimensional case has not been clarified. In this paper, we give the stationary amplitude for quantum walks on the d-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks. In fact, the amplitude for the stationary measure is an eigenfunction with eigenvalue 1.

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Acknowledgements

This work is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443).

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan

    Takashi Komatsu & Norio Konno

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  1. Takashi Komatsu
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  2. Norio Konno
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Correspondence to Takashi Komatsu.

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Komatsu, T., Konno, N. Stationary amplitudes of quantum walks on the higher-dimensional integer lattice. Quantum Inf Process 16, 291 (2017). https://doi.org/10.1007/s11128-017-1737-1

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